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Using law of sines prove that for any triangle in which either $B$ or $C$ is obtuse $$a = b\cos C + c\cos B$$

Use the law of sines to prove the additive property $$\sin(B+C)= \sin B\cos C + \cos B\sin C$$

These are the first questions in the book geometry revisited. I know I have to try all the questions myself but I have tried thinking about these questions for two days, but I couldn't find convincing questions.

I have proved the first one by using the hint given in The book. I don't t know why some fellow people put this question on hold. It involves some the concept of both trigonometry and geometry.

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    You should start with a picture. That usually helps in geometry. – 2017-01-07
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    Although googling may seem like a bit of a stretch the answer to both your questions will lie on the first page. – 2017-01-07
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    Do you need to prove the first one? – 2017-01-07
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    Do you understand what exactly David tries to say? Do you still want a full answer to help you? – 2017-01-07
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    I do not understand why on hold - the question is quite clear! Let us help the guy! – 2017-01-07
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    @Moti I have voted to reopen. I hope we get 4 more votes soon; I am looking forward to your answer. – 2017-01-07
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    "I have proved the first one by using the hint given in The book." Perhaps if you edit the question further to show your proof of this fact, the question might get the additional votes it needs to reopen. _Not_ including such details in the initial question posting may be a large part of the reason why the question was closed in the first place. (I cannot say for sure it was a reason, since I did not vote to close, but similar reasons are often given for closing other questions.) – 2017-01-09

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You have copied the exercise incorrectly from the book by Coxeter and Greitzer. (I assume you are using the 1967 edition of Geometry Revisited from the New Mathematical Library, an excellent series of small books published by the MAA.) The verbatim text of the exercise is:

  1. Show that, for any triangle $ABC,$ even if $B$ or $C$ is an obtuse angle, $a = b \cos C + c \cos B.$ Use the Law of Sines to deduce the "addition formula" $$ \sin (B+C) = \sin B \cos C + \sin C \cos B. $$

This exercise has two parts. The first is to show that $a = b \cos C + c \cos B.$ Contrary to what you wrote, the exercise does not say to use Law of Sines for this part. Indeed, I do not think it makes any sense to use the Law of Sines for this. Well-constructed figures with one or two added lines could do the job; I'd be tempted to use Cartesian coordinates in order to avoid the need to consider special cases. (Coordinates let you deal with negative values of cosines much like you deal with positive values.)

Also contrary to what you wrote, the exercise does not state that either $B$ or $C$ is obtuse; it merely says that one of them might be obtuse, that is, you cannot consider only the cases in which both $B$ and $C$ are acute or in which one of them is a right angle. In fact, you must consider all possible triangles. If assuming that $B$ is obtuse helps you prove the formula, that's fine; that covers the case where $B$ is obtuse, but then you must also somehow prove the case where $B$ is not obtuse.

After you show that $a = b \cos C + c \cos B,$ you are then to apply the Law of Sines in order to prove the angle addition formula for sines, using the already-proven fact that $a = b \cos C + c \cos B.$

You will make it easier for yourself if you copy the addition formula exactly as written in the book, rather than writing out the formula for $\sin(A+B)$ that you have probably seen somewhere else. The authors chose the particular symbols in their version of the formula in order to help you more easily use the Law of Sines to derive the addition formula from the equation $a = b \cos C + c \cos B.$ You can use the Law of Sines to make direct substitutions in that formula that get you almost all the way to the addition formula.

If the exercise had given the formula in the the form that you have written, the first step would be to realize that the formula is just as valid (or not valid) for $B$ and $C$ as for $A$ and $B,$ and rename the variables so that you are proving that $ \sin (B+C) = \sin B \cos C + \sin C \cos B. $

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    The question is CLEAR - B or C obtuse. The sine law clearly applies to the question in general and should be used to move from the first part (demonstrated with a simple picture) to the second. – 2017-01-07
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    @Moti The Sine Law is for moving from the first part to the second--yes, that's exactly what I said. But "even if $P$" does not imply $P$; it is merely a reminder that _this case also_ must be covered. Some students looking at such a problem might do only the acute-triangle case and need the reminder that not all triangles are acute. (The question as posted above _does_ say either $B$ or $C$ must be obtuse, but OP also cites the book. I supposed that OP wanted to solve the problem in the book, not some different problem.) – 2017-01-07
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    I am also a qualified teacher for math (and technology) and as such would focus on the fact that this is an exercise for "The Sine Law" where the first part does not need it (I think that this is implied from the fact that it is emphasized to be used for the second). Your lengthy response could easily intimidate every novice starter where I believe that we should try make math fan action. There is no need to mention acute since it is regarded as an "obvious" case - thus the emphasize on obtuse - no need to expand the discussion for such simple basic exercise. – 2017-01-07
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    @Moti Have you looked at the book? This truly is the first exercise. Only one proof (for the Sine Law itself) had previously been given, which (by the way) did explicitly deal with acute and obtuse triangles as separate cases. I do admit that my answer may be longer than it should be and may not be the most pedagogically sound. I would recommend you post an answer (it would likely be better), but unfortunately the question is on hold. – 2017-01-07
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    Let's hope we get the required votes. – 2017-01-09
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    +1 both because this is a good answer -- indeed, a model answer -- and because I disagree entirely with Moti's comments. – 2017-01-09
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    The guy asked about the obtuse case - so this should be the focus of the answer. We should assume that he understands the acute case. The simple and concise first comment by Simple is right - helping the guy understanding the nature of Cosine as a negative value for Obtuse angels. I will be happy to suggest a different approach to help the guy "learn" something from the requested kelp and not being intimidated by what I believe is way too complicated answer. When answers will be enable I will try to post my suggestion - which will contain graphics and specific hint. – 2017-01-09
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    @David it seems we are not successful... There is a really elegant and simple solution that I would love to hint to it... – 2017-01-11